## Experimental demonstration of the optical Zeno effect by scanning tunneling optical microscopy

Optics Express, Vol. 16, Issue 6, pp. 3762-3767 (2008)

http://dx.doi.org/10.1364/OE.16.003762

Acrobat PDF (285 KB)

### Abstract

An experimental demonstration of a classical analogue of the quantum Zeno effect for light waves propagating in engineered arrays of tunneling-coupled optical waveguides is reported. Quantitative mapping of the flow of light, based on scanning tunneling optical microscopy, clearly demonstrates that the escape dynamics of light in an optical waveguide side-coupled to a tight-binding continuum is slowed down when projective measurements, mimicked by sequential interruptions of the decay, are performed on the system.

© 2008 Optical Society of America

## 1. Introduction

1. B. Misra and E. C. G. Sudarshan, “The Zeno’s Paradox in Quantum Theory,” J. Math. Phys. **18**, 756–763 (1977). [CrossRef]

2. W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, “Quantum Zeno Effect,” Phys. Rev. A **41**, 2295–2300 (1990). [CrossRef] [PubMed]

3. P. Knight, “Watching a Laser Hot-Pot,” Nature (London) **344**, 493–494 (1990). [CrossRef]

4. H. Nakazato, M. Namiki, and S. Pascazio, “Temporal behavior of Quantum Mechanical Systems,” Int. J. Mod. Phys. B **10**, 247–295 (1996). [CrossRef]

5. A.G. Kofman and G. Kurizki, “Quantum Zeno effect on atomic excitation decay in resonators,” Phys. Rev. A **54**, R3750–R3753 (1996). [CrossRef] [PubMed]

6. M. Lewenstein and K. Rza̧zewski, “Quantum Anti-Zeno Effect,” Phys. Rev. A **61**, 022105-1-022105-5 (2000). [CrossRef]

7. A.G. Kofman and G. Kurizki, “Acceleration of Quantum Decay processes by frequent observations,” Nature (London) **405**, 546–550 (2000). [CrossRef] [PubMed]

8. A.G. Kofman and G. Kurizki, “Universal Dynamical Control of Quantum Mechanical Decay: Modulation of the Coupling to the Continuum,” Phys. Rev. Lett.87, 270405-1-270405-4 (2001). [CrossRef]

9. P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. **86**, 2699–2703 (2001). [CrossRef] [PubMed]

2. W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, “Quantum Zeno Effect,” Phys. Rev. A **41**, 2295–2300 (1990). [CrossRef] [PubMed]

11. E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, “Continuous and Pulsed Quantum Zeno Effect,” Phys. Rev. Lett. **97**, 260402-1-260402-4 (2006). [CrossRef]

11. E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, “Continuous and Pulsed Quantum Zeno Effect,” Phys. Rev. Lett. **97**, 260402-1-260402-4 (2006). [CrossRef]

12. P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. **89**, 080401-1-080401-4 (2002). [CrossRef] [PubMed]

14. A. Peres, “Zeno Paradox in Quantum Theory,” Am J. Phys.48, 931–932 (1980); G.S. Agarwal and S. P. Tewari, “An all-optical realization of quantum zeno effect,” Phys. Lett. A 185, 139–142 (1994); M. Kitano, “Quantum Zeno Effect and intracavity polarization filters,” Opt. Commun. 141, 39–42 (1997); V. Kidambi, A. Widom, C. Lerner, and Y. N. Srivastava, “Photon polarization measurements without the Quantum Zeno Effect,” Am. J. Phys. 68, 475–481 (2000); K. Yamane, M. Ito, and M. Kitano, “Quantum Zeno Effect in Optical Fibers,” Opt. Commun. 192, 299–307 (2001). [CrossRef]

7. A.G. Kofman and G. Kurizki, “Acceleration of Quantum Decay processes by frequent observations,” Nature (London) **405**, 546–550 (2000). [CrossRef] [PubMed]

15. M.C. Fischer, B. Gutierrez-Medina, and M.G. Raizen, “Observation of the Quantum Zeno and Anti-Zeno Effects in an unstable system,” Phys. Rev. Lett.87, 040402-1-040402-4 (2001). [CrossRef] [PubMed]

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

18. S. Longhi, “Control of Photon Tunneling in Optical Waveguides,” Opt. Lett.32, 557–559 (2007); “Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model,” Phys. Rev. B 75, 184306-1-184306-12 (2007); “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A 74, 063826-1-063826-14 (2006). [CrossRef] [PubMed]

6. M. Lewenstein and K. Rza̧zewski, “Quantum Anti-Zeno Effect,” Phys. Rev. A **61**, 022105-1-022105-5 (2000). [CrossRef]

7. A.G. Kofman and G. Kurizki, “Acceleration of Quantum Decay processes by frequent observations,” Nature (London) **405**, 546–550 (2000). [CrossRef] [PubMed]

8. A.G. Kofman and G. Kurizki, “Universal Dynamical Control of Quantum Mechanical Decay: Modulation of the Coupling to the Continuum,” Phys. Rev. Lett.87, 270405-1-270405-4 (2001). [CrossRef]

9. P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. **86**, 2699–2703 (2001). [CrossRef] [PubMed]

19. I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, “Quantum Zeno and anti-Zeno effects in the Friedrichs model,” Phys. Rev. A **63**, 062110-1-062110-10 (2001). [CrossRef]

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

20. G. Della Valle, S. Longhi, P. Laporta, P. Biagioni, L. Duò, and M. Finazzi, “Discrete diffraction in waveguide arrays: A quantitative analysis by tunneling optical microscopy,” Appl. Phys. Lett. **90**, 261118-1-261118-3 (2007). [CrossRef]

21. S.I. Bozhevolnyi and L. Kuipers, “Near-field characterization of photonic crystal waveguides,” Semicond. Sci. Technol. **21**, R1–R16 (2006). [CrossRef]

22. D.N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

## 2. Sample and methods

## 3. Model

*z*of a weakly waveguiding structure is described by a Schrödinger-like equation for the electric field amplitude

*Ψ*(

*x*,

*y*,

*z*), which exactly mimics the temporal evolution of a quantum particle in a two-dimensional multiple-well potential

*V*(

*x*,

*y*)≃

*n*-

_{s}*n*(

*x*,

*y*), where

*n*(

*x*,

*y*) is the refractive index profile of the waveguide-array system and

*n*the substrate refractive index. By substituting

_{s}*ħ*↦

*λ*/2

*π*,

*m*↦

*n*(

_{s}*m*being the mass of the particle in the Schrödinger equation), and

*t*↦

*z*, the temporal evolution of the particle wave function in the quantum problem is replaced, in our photonic structure, by the spatial propagation of the electric field amplitude

*Ψ*(

*x*,

*y*,

*z*) along

*z*[17

17. S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]

*P*to find the particle in the well W decays with time toward zero in absence of bound surface states [17

*χ*〉 into a tight-binding continuum |

*ω*〉 of states is commonly described in terms of the Friedrichs-Lee Hamiltonian

*H*=

*H*

_{0}+

*H*(see e.g. [7

_{I}**405**, 546–550 (2000). [CrossRef] [PubMed]

8. A.G. Kofman and G. Kurizki, “Universal Dynamical Control of Quantum Mechanical Decay: Modulation of the Coupling to the Continuum,” Phys. Rev. Lett.87, 270405-1-270405-4 (2001). [CrossRef]

9. P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. **86**, 2699–2703 (2001). [CrossRef] [PubMed]

*g*(

*ω*) is the discrete-continuum spectral coupling amplitude, 4

*ħ*Δ is the width of the continuum band, and

*σ*accounts for a possible detuning between the position of the discrete level and the center of the continuum, as shown in Fig. 1(d). For the system of our interest, the spectral coupling amplitude takes the specific form [18

18. S. Longhi, “Control of Photon Tunneling in Optical Waveguides,” Opt. Lett.32, 557–559 (2007); “Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model,” Phys. Rev. B 75, 184306-1-184306-12 (2007); “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A 74, 063826-1-063826-14 (2006). [CrossRef] [PubMed]

_{0}measures the strength of the discrete-continuum coupling.

22. D.N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

*c*(

_{χ}*z*) the complex amplitude of the modal field trapped in the waveguideW, the decay dynamics of

*c*(

_{χ}*z*) for the structure of Fig. 1(a) is described by the same Friedrichs-Lee Hamiltonian, where Δ

_{0}assumes the meaning of the coupling rate between the boundary waveguide W and the semi-array, Δ is the coupling rate between adjacent waveguides in the semi-array, and

*σ*accounts for a possible propagation constant mismatch of the mode in waveguideW as compared to waveguides in the semi-array. All these quantities are therefore expressed in units of mm

^{−1}in the optical realm. The decay evolution of

*c*(

_{χ}*z*) is obtained by standard spectral analysis and reads (see e.g. [9

**86**, 2699–2703 (2001). [CrossRef] [PubMed]

18. S. Longhi, “Control of Photon Tunneling in Optical Waveguides,” Opt. Lett.32, 557–559 (2007); “Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model,” Phys. Rev. B 75, 184306-1-184306-12 (2007); “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A 74, 063826-1-063826-14 (2006). [CrossRef] [PubMed]

*σ*|<2-(Δ

_{0}/Δ)

^{2}, there are no bound states and

*c*(

_{χ}*z*) (and hence the survival probability

*P*(

*z*) = |

*c*(

_{χ}*z*)|

^{2}) asymptotically decays to zero. However, the decay law shows non-exponential features which can be markedly pronounced [17

*σ*/Δ≃0. For the waveguide separations of the fabricated structure (

*a*= 9.5

*µ*m and

*a*

_{0}= 11

*µ*m), the values of the coupling rates between the waveguides turn out to be Δ≃0.435 mm

^{−1}and Δ

_{0}≃0.223 mm

^{−1}, as measured using two reference directional couplers (see e.g. [25

25. A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, “Control of directional evanescent coupling in fs laser written waveguides,” Opt. Express **15**, 1579–1587 (2007). [CrossRef] [PubMed]

*P*(

*z*) in the waveguide W of Fig. 1(a), as obtained by Eq. (4). The inset in Fig. 2 shows the corresponding behavior of the effective decay rate

*γ*

_{eff}(

*z*)=-(1/

*z*)ln

*P*(

*z*), compared to the ‘natural’ constant decay rate

**86**, 2699–2703 (2001). [CrossRef] [PubMed]

*P*(

*z*). The spatial length

*τ*= 3.1mm between successive interruptions in the sample of Fig. 1(b) plays the role of the time interval between successive measurements, and for the observation of the Zeno effect it should be chosen smaller than a characteristic ‘Zeno time’

*τ*

_{Z}, where

*τ*

_{Z}is the smallest root of the equation

*γ*

_{eff}(

*τ*

_{Z})=

*γ*

_{0}[9

**86**, 2699–2703 (2001). [CrossRef] [PubMed]

*τ*

_{Z}≃ 30 mm, i.e. much longer than the sample length (~12 mm), and anti-Zeno effect can be a priori excluded. Dashed line in Fig. 2 shows the same decay behavior simulated by means of a fully numerical integration of the paraxial wave equation for

*Ψ*(

*x*,

*y*,

*z*) using a beam-propagation software [26].

*P*(

*z*) is calculated by projecting, at each propagation distance

*z*, the envelope

*Ψ*(

*x*,

*y*,

*z*) over the fundamental mode of waveguide W. The good agreement between the two simulations further supports the use of a tight-binding approach (coupled-mode equations) to tackle the optical problem.

## 4. Results and discussion

*Ψ*(

*x*,0,

*z*)|

^{2}in the (

*x*,

*z*) plane of the sample are shown in Figs. 3(a) and 3(b) for the samples of Figs. 1(a) and 1(b), respectively.

*z*direction with steps of Δ

*z*= 500

*µ*m. Each scan covers a sample area of size 170

*µ*m×20

*µ*m in the (

*x*,

*z*) plane, thus comprising the main waveguideW and all the 16 waveguides in the semi-arrays S, S1, S2, etc. From the acquired STOM images, the integrated optical signal along the x direction is normalized in order to take absorption and internal losses into account. In this way the decay law for the light power trapped in waveguide W can be calculated, as fully described in [20

20. G. Della Valle, S. Longhi, P. Laporta, P. Biagioni, L. Duò, and M. Finazzi, “Discrete diffraction in waveguide arrays: A quantitative analysis by tunneling optical microscopy,” Appl. Phys. Lett. **90**, 261118-1-261118-3 (2007). [CrossRef]

*τ*between successive interruptions is large enough to ensure that light trapped in the semi-array after each interruption is scattered out into the substrate in a short distance (~600

*µ*m). In this way, the ‘memory’ in the continuum is erased after each interruption, as shown in the inset of Fig. 3(c), where the normalized power decay law in each of the semi-arrays is plotted and superimposed to the theoretical prediction.

## 5. Conclusion

## References and links

1. | B. Misra and E. C. G. Sudarshan, “The Zeno’s Paradox in Quantum Theory,” J. Math. Phys. |

2. | W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, “Quantum Zeno Effect,” Phys. Rev. A |

3. | P. Knight, “Watching a Laser Hot-Pot,” Nature (London) |

4. | H. Nakazato, M. Namiki, and S. Pascazio, “Temporal behavior of Quantum Mechanical Systems,” Int. J. Mod. Phys. B |

5. | A.G. Kofman and G. Kurizki, “Quantum Zeno effect on atomic excitation decay in resonators,” Phys. Rev. A |

6. | M. Lewenstein and K. Rza̧zewski, “Quantum Anti-Zeno Effect,” Phys. Rev. A |

7. | A.G. Kofman and G. Kurizki, “Acceleration of Quantum Decay processes by frequent observations,” Nature (London) |

8. | A.G. Kofman and G. Kurizki, “Universal Dynamical Control of Quantum Mechanical Decay: Modulation of the Coupling to the Continuum,” Phys. Rev. Lett.87, 270405-1-270405-4 (2001). [CrossRef] |

9. | P. Facchi, H. Nakazato, and S. Pascazio, “From the Quantum Zeno to the Inverse Quantum Zeno Effect,” Phys. Rev. Lett. |

10. | P. Facchi and S. Pascazio, |

11. | E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, “Continuous and Pulsed Quantum Zeno Effect,” Phys. Rev. Lett. |

12. | P. Facchi and S. Pascazio, “Quantum Zeno Subspaces,” Phys. Rev. Lett. |

13. | K. Koshino and A. Shimizu, “Quantum Zeno Effect for exponentially decaying systems ,” Phys. Rev. Lett. |

14. | A. Peres, “Zeno Paradox in Quantum Theory,” Am J. Phys.48, 931–932 (1980); G.S. Agarwal and S. P. Tewari, “An all-optical realization of quantum zeno effect,” Phys. Lett. A 185, 139–142 (1994); M. Kitano, “Quantum Zeno Effect and intracavity polarization filters,” Opt. Commun. 141, 39–42 (1997); V. Kidambi, A. Widom, C. Lerner, and Y. N. Srivastava, “Photon polarization measurements without the Quantum Zeno Effect,” Am. J. Phys. 68, 475–481 (2000); K. Yamane, M. Ito, and M. Kitano, “Quantum Zeno Effect in Optical Fibers,” Opt. Commun. 192, 299–307 (2001). [CrossRef] |

15. | M.C. Fischer, B. Gutierrez-Medina, and M.G. Raizen, “Observation of the Quantum Zeno and Anti-Zeno Effects in an unstable system,” Phys. Rev. Lett.87, 040402-1-040402-4 (2001). [CrossRef] [PubMed] |

16. |
See, for instance: D. Dragoman and M. Dragoman, |

17. | S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect,” Phys. Rev. Lett.97, 110402-1-110402-4 (2006). [CrossRef] [PubMed] |

18. | S. Longhi, “Control of Photon Tunneling in Optical Waveguides,” Opt. Lett.32, 557–559 (2007); “Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model,” Phys. Rev. B 75, 184306-1-184306-12 (2007); “Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir,” Phys. Rev. A 74, 063826-1-063826-14 (2006). [CrossRef] [PubMed] |

19. | I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, “Quantum Zeno and anti-Zeno effects in the Friedrichs model,” Phys. Rev. A |

20. | G. Della Valle, S. Longhi, P. Laporta, P. Biagioni, L. Duò, and M. Finazzi, “Discrete diffraction in waveguide arrays: A quantitative analysis by tunneling optical microscopy,” Appl. Phys. Lett. |

21. | S.I. Bozhevolnyi and L. Kuipers, “Near-field characterization of photonic crystal waveguides,” Semicond. Sci. Technol. |

22. | D.N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

23. | G. Della Valle, S. Taccheo, P. Laporta, G. Sorbello, E. Cianci, and V. Foglietti, “Compact high gain erbium-ytterbium doped waveguide amplifier fabricated by Ag-Na ion exchange,” Electron. Lett. |

24. | AlphaSNOM, WITec GmbH, Ulm, Germany. |

25. | A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tünnermann, “Control of directional evanescent coupling in fs laser written waveguides,” Opt. Express |

26. | BeamPROP, 5.0 ed., Rsoft Design Group, Inc., 2002. |

**OCIS Codes**

(000.1600) General : Classical and quantum physics

(080.1238) Geometric optics : Array waveguide devices

(180.4243) Microscopy : Near-field microscopy

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Microscopy

**History**

Original Manuscript: February 1, 2008

Revised Manuscript: March 1, 2008

Manuscript Accepted: March 2, 2008

Published: March 6, 2008

**Citation**

P. Biagioni, G. Della Valle, M. Ornigotti, M. Finazzi, L. Duò, P. Laporta, and S. Longhi, "Experimental demonstration of the optical Zeno effect by scanning tunneling optical microscopy," Opt. Express **16**, 3762-3767 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-3762

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### References

- B. Misra and E. C. G. Sudarshan, "The Zeno’s Paradox in Quantum Theory," J. Math. Phys. 18, 756-763 (1977). [CrossRef]
- W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, "Quantum Zeno Effect," Phys. Rev. A 41, 2295-2300 (1990). [CrossRef] [PubMed]
- P. Knight, "Watching a Laser Hot-Pot," Nature (London) 344, 493-494 (1990). [CrossRef]
- H. Nakazato, M. Namiki, and S. Pascazio, "Temporal behavior of Quantum Mechanical Systems," Int. J. Mod. Phys. B 10, 247-295 (1996). [CrossRef]
- A. G. Kofman and G. Kurizki, "Quantum Zeno effect on atomic excitation decay in resonators," Phys. Rev. A 54, R3750-R3753 (1996). [CrossRef] [PubMed]
- M. Lewenstein and K. Rz?zewski, "Quantum Anti-Zeno Effect," Phys. Rev. A 61, 022105-1-022105-5 (2000). [CrossRef]
- A.G. Kofman and G. Kurizki, "Acceleration of Quantum Decay processes by frequent observations," Nature (London) 405, 546-550 (2000). [CrossRef] [PubMed]
- A.G. Kofman and G. Kurizki, "Universal Dynamical Control of Quantum Mechanical Decay: Modulation of the Coupling to the Continuum," Phys. Rev. Lett. 87, 270405-1-270405-4 (2001). [CrossRef]
- P. Facchi, H. Nakazato, and S. Pascazio, "From the Quantum Zeno to the Inverse Quantum Zeno Effect," Phys. Rev. Lett. 86, 2699-2703 (2001). [CrossRef] [PubMed]
- P. Facchi and S. Pascazio, Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2001), Vol. 42, p. 147.
- E. W. Streed, J. Mun, M. Boyd, G. K. Campbell, P. Medley, W. Ketterle, and D. E. Pritchard, "Continuous and Pulsed Quantum Zeno Effect," Phys. Rev. Lett. 97, 260402-1-260402-4 (2006). [CrossRef]
- P. Facchi and S. Pascazio, "Quantum Zeno Subspaces," Phys. Rev. Lett. 89, 080401-1-080401-4 (2002). [CrossRef] [PubMed]
- K. Koshino and A. Shimizu, "Quantum Zeno Effect for exponentially decaying systems," Phys. Rev. Lett. 92, 030401-1-030401-4 (2004).
- A. Peres, "Zeno Paradox in Quantum Theory," Am J. Phys. 48, 931-932 (1980);G. S. Agarwal and S. P. Tewari, "An all-optical realization of quantum zeno effect," Phys. Lett. A 185, 139-142 (1994);M. Kitano, "Quantum Zeno Effect and intracavity polarization filters," Opt. Commun. 141, 39-42 (1997); V. Kidambi, A. Widom, C. Lerner, and Y. N. Srivastava, "Photon polarization measurements without the Quantum Zeno Effect," Am. J. Phys. 68, 475-481 (2000); K. Yamane, M. Ito, and M. Kitano, "Quantum Zeno Effect in Optical Fibers," Opt. Commun. 192, 299-307 (2001). [CrossRef]
- M. C. Fischer, B. Gutierrez-Medina, and M. G. Raizen, "Observation of the Quantum Zeno and Anti-Zeno Effects in an unstable system," Phys. Rev. Lett. 87, 040402-1-040402-4 (2001). [CrossRef] [PubMed]
- See, for instance: D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, Berlin, 2004) and references therein.
- S. Longhi, "Nonexponential decay via tunneling in tight-binding lattices and the optical zeno effect," Phys. Rev. Lett. 97, 110402-1-110402-4 (2006). [CrossRef] [PubMed]
- S. Longhi, "Control of Photon Tunneling in Optical Waveguides," Opt. Lett. 32, 557-559 (2007); "Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model," Phys. Rev. B 75, 184306-1-184306-12(2007); "Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir," Phys. Rev. A 74, 063826-1-063826-14 (2006). [CrossRef] [PubMed]
- I. Antoniou, E. Karpov, G. Pronko, and E. Yarevsky, "Quantum Zeno and anti-Zeno effects in the Friedrichs model," Phys. Rev. A 63, 062110-1-062110-10 (2001). [CrossRef]
- G. Della Valle, S. Longhi, P. Laporta, P. Biagioni, L. Duo, and M. Finazzi, "Discrete diffraction in waveguide arrays: A quantitative analysis by tunneling optical microscopy," Appl. Phys. Lett. 90, 261118-1-261118-3 (2007). [CrossRef]
- S. I. Bozhevolnyi and L. Kuipers, "Near-field characterization of photonic crystal waveguides," Semicond. Sci. Technol. 21, R1-R16 (2006). [CrossRef]
- D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
- G. Della Valle, S. Taccheo, P. Laporta, G. Sorbello, E. Cianci, V. Foglietti, "Compact high gain erbium-ytterbium doped waveguide amplifier fabricated by Ag-Na ion exchange," Electron. Lett. 42, 632-633 (2006). [CrossRef]
- AlphaSNOM, WITec GmbH, Ulm, Germany.
- A. Szameit, F. Dreisow, T. Pertsch, S. Nolte, and A. Tunnermann, "Control of directional evanescent coupling in fs laser written waveguides," Opt. Express 15, 1579-1587 (2007). [CrossRef] [PubMed]
- BeamPROP, 5.0 ed., Rsoft Design Group, Inc., 2002.

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